Translating a Word Problem into a System of Equations Exercises

Example 1

Set up a system of equations describing the following word problem. Be sure to specify what each variable represents.

A forest inhabited only by borogoves and mome raths contains eighty-two creatures. Borogoves are two-legged animals, while a mome rath has 4 legs. Sorry, we were getting bored of cats and birds.

If the forest contains a total of 234 legs, how many borogoves and how many mome raths live in the forest?

Answer

Let x be the number of borogoves and y the number of mome raths. Then

Example 2

Set up a system of equations describing the following word problem. Be sure to specify what each variable represents.

Stanley has $1.45 in dimes and quarters. If he has 10 coins total, how many of each kind of coin does he have?

Hint

You don't need to know the Canadian exchange rate to solve this problem.

Answer

Let x be the number of dimes and y the number of quarters Stanley has. Then

Example 3

Set up a system of equations describing the following word problem. Be sure to specify what each variable represents.

A private school spent $860,000 in one year to pay all of its teachers and administrators. The school has a total of twenty-four teachers and administrators combined. If teachers make $35,000 per year and administrators make $40,000 per year, how many teachers and how many administrators does the school have? Also, where in the country do these people live that they are able to get by on such a pittance? You hear that, private school? We're calling you out.

Answer

Let x be the number of teachers and y the number of administrators the school has. Then

Example 4

Set up a system of equations describing the following word problem. Be sure to specify what each variable represents.

Tom and Jerry drove 270 total miles in 5 hours. Tom always drives 40 mph and Jerry always drives 60 mph. It's a wonder they can manage to remain friends. How many hours did Tom drive and how many hours did Jerry drive?

Answer

Let x be the number of hours Tom drove and y the number of hours Jerry drove. Then

Example 5

Set up a system of equations describing the following word problem. Be sure to specify what each variable represents.

Blue beads cost $0.50 per ounce and green beads cost $0.75 per ounce. Janine wants to mix blue and green beads to get 10 ounces of a bead mixture worth $0.55 per ounce. Wow...Janine has some extremely specific designs on these beads, it seems. Hey, good for her for knowing exactly what she wants.

How many ounces of each color does Janine need to buy?

Answer

If x is the number of ounces of blue beads and y the number of ounces of green beads, then

Example 6

For each problem above, you found a system of equations. What did these systems of equations have in common?

Answer

In each system, one of the equations was

x + y = (some number).

In other words, one of the pieces of information given in the problem was the sum, or total, of the two unknown quantities. We're having sum fun now.

Example 7

Set up a system of equations describing the following word problem. Be sure to specify what each variable represents.

Ayako spent $7.80 on equal weights of white and red beads. The red beads cost twice as much per ounce as the white beads. Well, sure they do. The world's supply of the color red has recently dipped to an all-time low, and is in high demand.

How much did Ayako spend on each color of beads?

Answer

Let x be the amount Ayako spent on white beads, and y the amount she spent on red beads. Then

Example 8

Set up a system of equations describing the following word problem. Be sure to specify what each variable represents.

A rectangle has perimeter 18. If each side of the rectangle is doubled, the perimeter of the new rectangle is 36. What were the dimensions of the original rectangle?

Answer

Let l be the length and w the width of the original rectangle. Then

Example 9

Set up a system of equations describing the following word problem. Be sure to specify what each variable represents.

5 ounces of dark chocolate and 11 ounces of milk chocolate combine to form one pound of 52.5% chocolate. Eight ounces each of dark and milk chocolate combine to form one pound of 60% chocolate. What percent chocolate are the dark and milk chocolate being used? Bonus question: dark chocolate kicks milk chocolate's behind. Okay, so the bonus question wasn't actually a question, but now you know where we stand.

Answer

Let x be the percent of the dark chocolate and y the percent of the milk chocolate, each represented as a number between 0 and 1. Then

Example 10

Set up a system of equations describing the following word problem. Be sure to specify what each variable represents.

Jemima wants to make chocolate-chip walnut brownies. Chocolate chips come in a 12 oz bag that costs $3. Walnuts come in a 4 oz bag that costs $2. If Jemima needs three pounds of chocolate chips and walnuts combined, and has $15 to spend, how many bags of each can she buy? BTW, what's with the shoestring budget, Jemima? If you're going to start this chocolate-chip walnut brownie venture of yours, build up some capital and really go for it. Even if you're only entertaining a few guests.

Answer

Let x be the number of bags of chocolate chips and y the number of bags of walnuts. There are 16 ounces in a pound, so to get the right weight,